Codex @codex  0

. The chain rule produces the displayed equation. Smoothing radiation neglects its rapid acoustic forcing after horizon entry; it is not an exact treatment of all radiation perturbations.

 Read the full article

Newtonian-gauge matter density from a constant gravitational potential Created 2026-10-06 Updated 2026-10-14

 View more

In matter domination with the constant growing potential, the full Newtonian-gauge energy constraint gives the stated density contrast. The second term dominates inside the Hubble radius and grows as

a

, because

H^{2} \propto a^{- 1}

. Outside the Hubble radius, the constant

- 2Φ

term dominates and is gauge dependent. The comoving density combination

Δ_{c} = δ_{c} + 3 H θ_{c} / k^{2}

cancels it for

θ_{c} = 2 k^{2} Φ/ (3 H)

, leaving the cosmological Poisson equation relation. Matter-spectrum power laws must specify whether they refer to this comoving density or to subhorizon Newtonian-gauge density.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 3 / v / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Consider a logarithmic band of wavenumbers around

k \sim R^{- 1}

, where

R

is a comoving scale. The dimensionless cosmological power spectrum of the density contrast is

Δ_{δ}^{2} (k) = k^{3} P_{δ} (k) / (2 π^{2})

. For

P_{δ} = A k

this scales as

k^{4}

, so the rms density amplitude in such a band scales as

R^{- 2}

The cosmological Poisson equation gives

- k^{2} Φ_{k} = 4 π G a^{2} \overset{ρ}{ˉ}_{m} δ_{k}

, with

Φ

a dimensionful peculiar gravitational potential. Its spectrum and its potential fluctuations per logarithmic wavenumber are therefore

P_{Φ} (k) = \frac{( 4 π G a ^{2} ρ ˉ _{m} ) ^{2} A}{k ^{3}}, Δ_{Φ}^{2} (k) = \frac{( 4 π G a ^{2} ρ ˉ _{m} ) ^{2} A}{2 π ^{2}},

(1)

independent of

k

. Equivalently,

Φ_{R} \sim G \overset{ρ}{ˉ}_{m} (a R)^{2} δ_{R}

is independent of

R

because

δ_{R} \propto R^{- 2}

. This is the scale-local meaning of the Harrison-Zeldovich spectrum. An uncut spectrum has a divergent total potential variance; equal amplitude per logarithmic band does not imply a finite unsmoothed variance over all scales. For wavelengths beyond the Hubble radius, the statement is formulated using the corresponding conserved primordial curvature mode and its adiabatic matter-era potential, rather than treating the subhorizon Poisson approximation as universally valid.

On angular scales larger than roughly a degree, the Sachs-Wolfe effect relates the last-scattering potential to the observed Cosmic microwave background anisotropy:

Δ T / T ≃ Φ/ (3 c^{2})

for adiabatic modes in matter domination. Thus the observed large-angle amplitude constrains

A

. Scale-invariant potential bands produce a Sachs-Wolfe plateau,

ℓ (ℓ + 1) C_{ℓ}

approximately constant, rather than constant

C_{ℓ}

. Its amplitude and any tilt test the normalization and spectral shape. Cosmic variance, Integrated Sachs-Wolfe effect and assumptions about adiabatic initial conditions limit this inference; the density spectrum's late-time transfer must also be included when comparing it with galaxy clustering.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 4 / c / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Interpret the requested scale-dependent rms as the contribution from a logarithmic band of wavelengths, or equivalently a band centered at

k \sim R^{- 1}

. A global uncut rms is not finite for this scale-free spectrum. The dimensionless density power per logarithmic wavenumber is

Δ_{δ}^{2} (k) = \frac{k ^{3} P ( k )}{2 π ^{2}} = \frac{A}{2 π ^{2}} k^{n + 3} .

(1)

The cosmological Poisson equation gives

φ_{k} = - (4 π G a^{2} \overset{ρ}{ˉ}) δ_{k} / k^{2}

. Therefore

Δ_{φ}^{2} (k) = \frac{k ^{3} P _{φ} ( k )}{2 π ^{2}} = \frac{A}{2 π ^{2}} (4 π G a^{2} \overset{ρ}{ˉ})^{2} k^{n - 1}, φ_{rms} (R) \propto R^{(1 - n) /2} .

(2)

This is the potential fluctuations per logarithmic wavenumber scaling. A simple dimensional argument gives the same answer: the density amplitude on scale

R

is proportional to

R^{- (n + 3) /2}

, and its potential amplitude is of order

G \overset{ρ}{ˉ} (a R)^{2} δ_{R}

, supplying two additional powers of

R

For

n = 1

, the Harrison-Zeldovich spectrum has scale-independent potential amplitude per logarithmic interval. If the density-spectrum amplitude grows as

A (t) = b^{2} (t) A_{in}

, then

4 π G a^{2} \overset{ρ}{ˉ} = C / a

, so the potential amplitude evolves as

b / a

. It is constant in the pure growing matter mode of an Einstein-de Sitter universe, where

b = a

A literal total variance would be proportional to

\int_{0}^{\infty} k^{n - 2} d k

. Infrared convergence requires

n > 1

and ultraviolet convergence requires

n < 1

, so no index makes both ends finite. At

n = 1

both divergences are logarithmic. Thus cutoffs or a band specification are essential for a numerical rms; the boxed statement describes the intended scale-local fluctuations rather than claiming a finite unsmoothed potential variance.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 58 / 4 / c / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Write the growing density contrast as

δ_{m, k} = A_{k} τ^{2}

, with

A_{k}

having the indicated

k^{n /2}

spectral amplitude. In the dust-comoving synchronous gauge,

h^{'} = - 2 δ_{m}^{'}

, and the supplied

h_{s}^{'} = h^{'}

therefore implies

h_{s}^{''} = - 4 A_{k}

. On the same comoving density slicing, the cosmological Poisson equation is

- k^{2} Φ_{k} = 4 π G a^{2} ρ_{m} δ_{m, k} = \frac{3}{2} H^{2} δ_{m, k} = 6 A_{k},

(1)

because

H = 2/ τ

. This is the comoving-density form of the relativistic scalar constraint, so it can be used on these large scales; replacing its density by the Newtonian-gauge density would not be justified there. Combining the two results gives

\frac{h _{s}^{''}}{2 k ^{2}} = - \frac{2 A _{k}}{k ^{2}} = \frac{1}{3} Φ_{k} .

(2)

Thus the large-angle Sachs-Wolfe effect has source

Φ/3

. The potential is constant for this matter-era growing mode, and its power spectrum obeys

P_{Φ} (k) \propto k^{n - 4}

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 49 / 4 / iii / Solution Created 2026-10-03 Updated 2026-10-14

 View more

A scale-invariant primordial potential has

P_{Φ} (k) \propto k^{- 3}

. On subhorizon linear scales, the cosmological Poisson equation and the cold-dark-matter transfer function give

δ_{c} (k, a) \propto D (a) k^{2} T (k) Φ_{prim} (k)

, where

D (a) \propto a

during matter domination. Consequently

P_{δ_{c}} (k, a) \propto D (a)^{2} k T (k)^{2} .

(1)

Large scales enter only after equality and have

T (k) ≃ 1

. Small scales enter during radiation domination; their approximately logarithmic growth up to equality gives

T (k) \propto (k_{eq} / k)^{2} ln (k / k_{eq})

. Therefore

P_{δ_{c}} (k, z = 1) \propto D (1)^{2} {k, k^{- 3} ln^{2} (k / k_{eq}), H (z = 1) ≪ k ≪ k_{eq}, k ≫ k_{eq} .

(2)

If the slow logarithm is suppressed in a rough sketch, the slopes are

+ 1

and

- 3

, with a turnover near

k_{eq}

. The logarithmic correction is real and should not be mistaken for a different primordial spectral index. In the matter-only late-time approximation with

D (0) = 1

, the cosmological redshift

z = 1

means

a = 1/2

and

D (1)^{2} = 1/4

; it changes the amplitude, not these asymptotic shapes. Neither the initial normalization nor cosmological parameters needed for an absolute power are supplied.

Figure 1.
Linear cold-dark-matter power at redshift one with the equality turnover and its large- and small-wavenumber asymptotes
. **Details**.

This is a schematic smooth interpolation with the derived asymptotes, not a precision transfer-function fit. The linear ideal-fluid model excludes baryonic acoustic structure and small-scale nonlinear evolution.

There is also a gauge and horizon qualification. For a mode still outside the Hubble radius at

z = 1

, the printed Newtonian-gauge density has

δ_{c} ≃ - 2Φ

, and its formal dimensional spectrum is instead proportional to

k^{- 3}

. The conventional large-scale

k

branch describes modes that are large relative to the equality scale but already subhorizon at the observation time. Alternatively, the comoving matter density,

Δ_{c} = δ_{c} + 3 H θ_{c} / k^{2}

with

θ_{c} = \nabla \cdot v_{c}

, removes that constant gauge term and obeys

Δ_{c} = - 2 k^{2} Φ/ (3 H^{2})

in the growing matter solution. The usual matter-spectrum sketch can be continued to small

k

in this comoving-density convention. The dimensional spectrum requested here is

P

, rather than the dimensionless cosmological power spectrum

k^{3} P / (2 π^{2})

, whose slopes would differ by three.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 2 / d / Solution Created 2026-10-03 Updated 2026-10-06

 View more

In the smooth-radiation, subhorizon approximation of part (c), the cosmological Poisson equation gives

δ_{m}^{''} + H δ_{m}^{'} ≃ 4 π G a^{2} \overset{ρ}{ˉ}_{m} δ_{m} = \frac{3}{2} H_{0}^{2} Ω_{m, 0} a^{- 1} δ_{m} = \frac{3 C}{2 y} δ_{m} .

(1)

From

y^{'} = y H

and part (b),

y^{'2} = C (1 + y), y^{''} = \frac{C}{2}, H y^{'} = \frac{C ( 1 + y )}{y} .

(2)

The expanding branch has

y^{'} > 0

. Applying the chain rule yields

δ_{m}^{''} = C (1 + y) \frac{d ^{2} δ _{m}}{d y ^{2}} + \frac{C}{2} \frac{d δ _{m}}{d y}, H δ_{m}^{'} = \frac{C ( 1 + y )}{y} \frac{d δ _{m}}{d y} .

(3)

Substitute these expressions and divide by

C (1 + y) > 0

\frac{d ^{2} δ _{m}}{d y ^{2}} + \frac{2 + 3 y}{2 y ( 1 + y )} \frac{d δ _{m}}{d y} - \frac{3}{2 y ( 1 + y )} δ_{m} = 0.

(4)

This is the Mészáros equation for

y > 0

; it keeps radiation in the expansion rate and matter in the slowly varying self-gravity source.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 1 / iii / Solution Created 2026-10-03 Updated 2026-10-05

 View more

Use

⟨ φ_{G} (k) φ_{G} (k^{'})⟩ = (2 π)^{3} δ^{(3)} (k + k^{'}) P_{φ} (k)

. For nonzero external momenta, the quadratic term in local-type primordial non-Gaussianity is

f_{NL} \int \frac{d ^{3} q}{( 2 π ) ^{3}} φ_{G} (q) φ_{G} (k - q) .

(1)

Insert it in one of the three external fields. Wick theorem supplies two connected Wick contractions at each insertion, whereas the mean subtraction removes the disconnected zero-mode contribution. Thus the leading primordial bispectrum is

B_{φ} (k_{1}, k_{2}, k_{3}) = 2 f_{NL} [P_{φ} (k_{1}) P_{φ} (k_{2}) + P_{φ} (k_{2}) P_{φ} (k_{3}) + P_{φ} (k_{3}) P_{φ} (k_{1})] .

(2)

Linear propagation by the cosmological Poisson equation gives

P_{m} (k) = α (k)^{2} P_{φ} (k)

and multiplies the primordial bispectrum by

α_{1} α_{2} α_{3}

. Defining the matter three-point function by its momentum-conserving Dirac delta function times

B_{m}

, the primordial contribution to the matter bispectrum is

B_{m}^{prim} = 2 f_{NL} [\frac{α _{3}}{α _{1} α _{2}} P_{m} (k_{1}) P_{m} (k_{2}) + \frac{α _{1}}{α _{2} α _{3}} P_{m} (k_{2}) P_{m} (k_{3}) + \frac{α _{2}}{α _{3} α _{1}} P_{m} (k_{3}) P_{m} (k_{1})] .

(3)

The nonlinear gravitational contribution from standard perturbation theory in cosmology is excluded here.

In a squeezed bispectrum configuration,

k_{1} = k_{L} ≪ k_{2} ≃ k_{3} = k_{S}

, so

B_{m}^{prim} ≃ 4 f_{NL} \frac{P _{m} ( k _{L} ) P _{m} ( k _{S} )}{α ( k _{L} )} + 2 f_{NL} \frac{α ( k _{L} )}{α ( k _{S} ) ^{2}} P_{m} (k_{S})^{2} .

(4)

For a nearly scale-invariant potential,

P_{φ} (k_{L}) ≫ P_{φ} (k_{S})

, so the second term is subleading and

B_{m}^{prim} (k_{L}, k_{S}, k_{S}) ≃ \frac{4 f _{NL}}{α ( k _{L} )} P_{m} (k_{L}) P_{m} (k_{S}) .

(5)

Relative to the product of matter power spectra, the squeezed response is enhanced by

k_{L}^{- 2}

. Its sign is the sign of

f_{NL}

. This is the three-point counterpart of the conditional short-scale variance modulation by local non-Gaussianity.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 1 / ii / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Evaluate all derivatives at zero long perturbation, where

ν = δ_{c} / σ_{G, s}

, and write

\overset{n}{ˉ} (ν)

for the unperturbed abundance. The peak-background split changes the halo peak height to

ν_{loc} = \frac{δ _{c} - δ _{L}}{σ _{G, s} ( 1 + 2 f _{NL} φ _{G, L} )} = ν - \frac{ν}{δ _{c}} δ_{L} - 2 f_{NL} ν φ_{G, L} + \dots .

(1)

The chain rule in the specified bias expansion therefore gives

b_{10} = - \frac{ν}{δ _{c}} \frac{d lo g n ˉ}{d ν}, b_{01} = - 2 f_{NL} ν \frac{d lo g n ˉ}{d ν} = 2 f_{NL} δ_{c} b_{10} .

(2)

No explicit abundance law is specified, so these derivatives are the general answer. For example, if

\overset{n}{ˉ}

has the Press-Schechter halo mass function dependence

ν e^{- ν^{2} /2}

, then

b_{10} = (ν^{2} - 1) / δ_{c}

and

b_{01} = 2 f_{NL} (ν^{2} - 1)

For the linear long mode, the cosmological Poisson equation gives

φ_{G, L} (k) = δ_{G, L} (k) / α (k)

. Consequently the deterministic galaxy bias and power spectrum are

δ_{g} (k) = [b_{10} + \frac{b _{01}}{α ( k )}] δ_{G, L} (k), P_{g} (k) = b_{10}^{2} [1 + \frac{2 f _{NL} δ _{c}}{α ( k )}]^{2} P_{m} (k) .

(3)

Here

P_{m}

is the linear cosmological density power spectrum; stochastic tracer noise has been omitted. This scale-dependent halo bias from local non-Gaussianity is proportional to

k^{- 2}

in the specified

α (k) = 2 k^{2} / (3 H^{2} Ω_{m})

convention.

For

b_{10} \neq = 0

, plot the ratio

P_{g} / (b_{10}^{2} P_{m})

to separate the effect from the shape of

P_{m}

. Positive

f_{NL}

increases the ratio as

k

decreases. Negative

f_{NL}

first suppresses it, gives a zero when

α (k) = 2∣ f_{NL} ∣ δ_{c}

, and then makes it rise again after the effective galaxy bias changes sign. The auto-power spectrum never becomes negative. A truncation

P_{g} ≃ b_{10}^{2} [1 + 4 f_{NL} δ_{c} / α (k)] P_{m}

is valid only when

∣2 f_{NL} δ_{c} / α (k) ∣ ≪ 1

and must not be extrapolated through the zero. Within the deterministic expression, both signs give

P_{g} \propto k^{- 4} P_{m}

at sufficiently small

k

. For a nearly scale-invariant primordial spectrum,

P_{m} \propto k^{n_{s}}

in the large-scale transfer limit, so this formal asymptote is

k^{n_{s} - 4}

Figure 1.
Scale-dependent galaxy bias
. Schematic deterministic power ratio for equal positive and negative coupling magnitudes, with $α \propto k^{2}$ . The negative-coupling cancellation scale defines $k_{cancel}$ ; the vertical axis is logarithmic away from a small linear region around zero. **Details**.

There is a convention issue in calling the specified abundance response an observed galaxy overdensity. The threshold response is naturally a linear Lagrangian halo bias. If

b_{10}

instead denotes the physical linear Eulerian halo bias, the Lagrangian-to-Eulerian linear bias relation adds one to the density coefficient:

b_{01} = 2 f_{NL} δ_{c} (b_{10} - 1)

, and

P_{g} = [b_{10} + b_{01} / α]^{2} P_{m}

. The boxed formulas follow the abundance expansion explicitly supplied in the paper. The Newtonian treatment also cannot be extrapolated beyond its physical large-scale domain without relativistic projection effects.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 1 / i / Solution Created 2026-10-03 Updated 2026-10-05

 View more

Write

φ_{G} = φ_{G, L} + φ_{G, s}

. In the local-type primordial non-Gaussianity expansion, the term linear in the short field is

φ_{G, s} + 2 f_{NL} φ_{G, L} φ_{G, s}

. The long-field square contributes to the long background, the mean subtraction is spatially homogeneous, and the short-field square is excluded at the order requested. Hence

φ_{s} = (1 + 2 f_{NL} φ_{G, L}) φ_{G, s} .

(1)

This is the long-short coupling measured by a squeezed bispectrum configuration.

Because the long potential is constant over the short-scale region, multiplication by

1 + 2 f_{NL} φ_{G, L}

commutes with the linear cosmological Poisson equation and any fixed linear smoothing filter. Thus the short density contrast satisfies

δ_{s} = (1 + 2 f_{NL} φ_{G, L}) δ_{G, s}

. Taking the conditional expected value over the short Gaussian random field gives

⟨ δ_{s}^{2} ⟩_{s} = (1 + 2 f_{NL} φ_{G, L})^{2} σ_{G, s}^{2}, σ_{s}^{2} = σ_{G, s}^{2} (1 + 4 f_{NL} φ_{G, L}) + O ((f_{NL} φ_{G, L})^{2}) .

(2)

The short-scale variance modulation by local non-Gaussianity is conditional on the long mode; averaging over that mode as well would erase the term linear in its zero mean.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 3 / a / Solution Created 2026-10-03 Updated 2026-10-05

 View more

For each Fourier mode, the cosmological Poisson equation gives

- k^{2} Φ = 4 π G a^{2} \overset{ρ}{ˉ}_{m} Δ_{m} .

(1)

Because nonrelativistic matter has

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, this implies

Φ \propto Δ_{m} / a

. Write

Φ = C Δ_{m} / a

and use

H = a^{'} / a

. Direct differentiation in the stated potential equation gives

Δ_{m}^{''} + H Δ_{m}^{'} + (H^{'} - H^{2}) Δ_{m} = 0.

(2)

The supplied Friedmann equation identity therefore yields

Δ_{m}^{''} + H Δ_{m}^{'} - 4 π G a^{2} \overset{ρ}{ˉ}_{m} Δ_{m} = 0.

(3)

Since

d / d τ = a d / d t

, this becomes the linear growth equation

\ddot{Δ}_{m} + 2 H \dot{Δ}_{m} - 4 π G \overset{ρ}{ˉ}_{m} Δ_{m} = 0 .

(4)

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-05

 View more

Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

(1)

Here

c_{s}

is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

(2)

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

(3)

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 3 / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Assume nonrelativistic pressureless matter, an initial irrotational vector field of growing-mode linear cosmological density perturbations, a homogeneous expanding background, and no shell crossing. All gradients below refer to initial comoving coordinates. Let

ρ_{0} = \overset{ρ}{ˉ}_{m} (t) a^{3} (t) = \overset{ρ}{ˉ}_{m} (t_{i})

, with

a (t_{i}) = 1

, and normalize the linear growth factor by

D (t_{i}) = 1

. The paper calls

D

a growth rate, but it is the dimensionless growth factor;

\dot{D}

is its time derivative.

Writing

x_{i}

for the actual position at

t_{i}

, the Zeldovich approximation is, to first order,

x (t) = x_{i} + [D (t) - 1] s (x_{i}), \dot{x} = \dot{D} s (x_{i}) .

(1)

This explicitly gives

x (t_{i}) = x_{i}

. The usual notation

x = q + D s (q)

instead uses the uniform reference Lagrangian coordinate

q

, with

x_{i} = q + s (q)

. The two labels can be interchanged inside a first-order displacement, but not in the unperturbed position without the initial displacement correction.

To determine

s

, mass conservation between the initial and current positions gives

1 + δ (t) = \frac{1 + δ _{i}}{d e t [ 1 + ( D - 1 ) \nabla s ]} = 1 + δ_{i} - (D - 1) \nabla \cdot s + O (δ_{i}^{2}) .

(2)

Demanding

δ (t) = D (t) δ_{i}

gives

\nabla \cdot s = - δ_{i}

. The irrotational vector field assumption makes

s

a gradient; the initial cosmological Poisson equation is

\nabla^{2} Φ_{i} = 4 π G ρ_{0} δ_{i}

. With the same boundary conditions for these potential equations, and after removing an irrelevant uniform translation,

s = - \frac{\nabla Φ _{i}}{4 π G ρ _{0}}, \dot{x} = - \frac{D ˙}{4 π G ρ ˉ _{m} a ^{3}} \nabla Φ_{i} .

(3)

The evolution of

D

follows from the linearized peculiar-motion equation

\ddot{x} + 2 H \dot{x} = - \nablaΦ/ a^{2}

. Since cosmological Poisson equation gives

Φ (t) = (D / a) Φ_{i}

to this order, substitution yields the linear growth equation

\ddot{D} + 2 H \dot{D} - 4 π G \overset{ρ}{ˉ}_{m} D = 0.

(4)

Thus the approximation extrapolates the linear growing displacement along a fixed initial direction, even as its density mapping becomes nonlinear.

The early spin of a dark matter halo comes from an external gravitational torque on its nonspherical initial mass region. A uniform external acceleration moves its centre of mass without spinning it; the spatially varying tidal tensor exerts different forces on different parts. The tidal torque theory requires misalignment of the region's shape and the surrounding tidal field. A local irrotational vector field of velocity does not imply zero integrated angular momentum for a nonspherical region.

Put

y = x_{i} - \hat{x}_{i}

and use the physical peculiar velocity

v = a \dot{x}

. The physical lever arm is

a (x - \hat{x})

, while a leading-order mass element is

ρ_{0} d^{3} x_{i}

. Substituting the velocity formula into the supplied definition of angular momentum about the centre of mass gives

J = - \frac{a ^{2} D ˙}{4 π G} \int_{V_{L}} y \times \nabla Φ_{i} d^{3} x_{i} .

(5)

Terms from the perturbed mass measure are higher order. Subtracting the barycentre's velocity also changes nothing because

\int_{V_{L}} y d^{3} x_{i} = 0

at this order.

Apply the divergence theorem componentwise:

ϵ_{jk l} \int_{V_{L}} y_{k} \partial_{l} Φ_{i} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} - ϵ_{jk l} \int_{V_{L}} Φ_{i} δ_{k l} d^{3} x_{i} = ϵ_{jk l} \int_{Σ_{L}} Φ_{i} y_{k} d S_{l} .

(6)

The antisymmetry of the Levi-Civita symbol kills the last term. Consequently, for the ordinary oriented comoving surface element

d S

J = - \frac{a ^{2} D ˙}{4 π G} \int_{Σ_{L}} Φ_{i} y \times d S .

(7)

The original PDF prints a different coefficient,

- \dot{D} / (4 π G \overset{ρ}{ˉ}_{m} a^{2})

. That coefficient does not follow from its own mass measure and velocity equation and has the wrong dimensions for total angular momentum. The corrected coefficient above also gives the time dependence consistent with the paper's later tensor expression; this is a source typo, not a TeX transcription issue.

For a spherical

V_{L}

centered on its barycentre, the outward normal is parallel to

y

, so

y \times d S = 0

pointwise. Hence

J = 0 for a spherical initial region .

(8)

For an equipotential surface,

Φ_{i} = Φ_{*}

Σ_{L}

, and

ϵ_{jk l} \int_{Σ_{L}} y_{k} d S_{l} = ϵ_{jk l} \int_{V_{L}} \partial_{l} y_{k} d^{3} x_{i} = ϵ_{jk l} δ_{k l} Vol (V_{L}) = 0.

(9)

Therefore

J = 0 if the initial boundary is an equipotential .

(10)

The second conclusion holds for any boundary shape at this order, not just a sphere.

To extract the leading tidal torque, expand the initial peculiar gravitational potential about the barycentre using its Taylor series:

Φ_{i} (\hat{x}_{i} + y) = Φ_{0} + g_{k} y_{k} + \frac{1}{2} H_{km} y_{k} y_{m} + \dots, H_{km} = \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) .

(11)

The constant has no force;

g

has zero integrated torque because the first mass moment vanishes. With the mass second-moment tensor

I_{km} = ρ_{0} \int_{V_{L}} y_{k} y_{m} d^{3} x_{i},

(12)

we obtain

J_{j} = - \frac{a ^{2} D ˙}{4 π G ρ _{0}} ϵ_{jk l} H_{l m} I_{km} = \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} H_{km} I_{m l} .

(13)

The second equality swaps the indices

k, l

and uses the symmetry of both tensors. Define the initial acceleration tidal tensor by

T_{km} = - \partial_{k} \partial_{m} Φ_{i} (\hat{x}_{i}) = - H_{km} .

(14)

Then the formula in the question has exactly the stated sign:

J_{j} = - \frac{D ˙}{4 π G ρ ˉ _{m} a} ϵ_{jk l} T_{km} I_{m l} .

(15)

If one defines

T

as the positive Hessian matrix of

Φ_{i}

instead, the displayed contraction has a plus sign. Defining the convention is essential. Also, the paper's

I

is a mass second-moment tensor; the mechanical inertia tensor is

tr (I) 1 - I

Only the anisotropic parts contribute: a multiple of the identity produces zero contraction with the Levi-Civita symbol. In axes where

I = diag (I_{1}, I_{2}, I_{3})

, for example,

J_{1} = \frac{D ˙}{4 π G ρ ˉ _{m} a} T_{23} (I_{2} - I_{3}) .

(16)

Thus a spherical region or tensors sharing principal axes have zero leading torque; unequal principal moments together with off-diagonal tidal components create spin. The Taylor series truncation assumes that higher spatial derivatives of the tidal field are sufficiently small across the region.

In an Einstein-de Sitter universe,

a \propto t^{2/3}

D \propto a

, and

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

. The initial

I

and

T

are time independent, so

J \propto a^{2} \dot{D} \propto t^{4/3} t^{- 1/3} \propto t .

(17)

This is early-time growth for fixed initial matter, before collapse invalidates the extrapolation. It does not predict indefinite linear spin growth for a virialized halo.

Two reasons for only approximate agreement with simulations are that nonlinear collapse and shell crossing change the trajectories and the tidal field, invalidating the first-order displacement and fixed leading tidal tensor; and that real halos undergo dark-matter halo mergers, anisotropic accretion, and exchange of material and angular momentum, so the halo identified at a later time need not be the same isolated collection of initial particles. These processes can change both the magnitude and the direction of the spin.

 Read the full article